CONDITIONAL ENTROPY: A Tool to Explore the Phase Space

P. CINCOTTA, Facultad de Ciencias Astron\'omicas y Geof\'{\i}sicas, 
   Universidad Nacional de La Plata, Paseo del Bosque, 1900 La Plata, Argentina,
   and
   Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona, 
   Gran Via 585, 08007 Barcelona, Spain 

C. SIM\'O, Departament de Matem\`atica Aplicada i An\`alisi, Universitat de
   Barcelona, Gran Via 585, 08007 Barcelona, Spain

Abstract

In this paper we show that the Conditional Entropy of nearby orbits may be
a useful tool to explore the phase space associated to a given Hamiltonian.
The arc length parameter along the orbits, instead of the time, is used as
a random variable to compute the entropy. In the first part of this work we
summarize the main analytical results to support this tool while, in the
second part, we present numerical evidence that this technique is able to 
localize (stable) periodic and quasiperiodic orbits, `aperiodic' orbits
(chaotic motion) and unstable periodic orbits (the `source' of chaotic 
motion). Besides, we show that this technique provides a measure of chaos
which is similar to that given by the Lyapunov Characteristic Number.
It is important to remark that this method is very simple to compute and
does not require long time integrations, just realistic physical times.